x^2 - 34x + c = 0 In the given equation, c is a constant. The equation has no real...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{x^2 - 34x + c = 0}\)
In the given equation, \(\mathrm{c}\) is a constant. The equation has no real solutions if \(\mathrm{c \gt n}\). What is the least possible value of \(\mathrm{n}\)?
1. TRANSLATE the problem information
- Given equation: \(\mathrm{x^2 - 34x + c = 0}\)
- The equation has no real solutions when \(\mathrm{c \gt n}\)
- We need to find the least possible value of n
2. INFER the mathematical approach
- "No real solutions" for a quadratic equation means the discriminant must be negative
- We need to find when the \(\mathrm{discriminant \lt 0}\), then determine the threshold value n
3. INFER the discriminant setup
- For quadratic \(\mathrm{ax^2 + bx + c = 0}\), \(\mathrm{discriminant = b^2 - 4ac}\)
- In our equation: \(\mathrm{a = 1, b = -34, c = c}\)
- Discriminant = \(\mathrm{(-34)^2 - 4(1)(c) = 1156 - 4c}\)
4. SIMPLIFY the inequality condition
- For no real solutions: \(\mathrm{1156 - 4c \lt 0}\)
- Add 4c to both sides: \(\mathrm{1156 \lt 4c}\)
- Divide by 4: \(\mathrm{289 \lt c}\)
- This means: \(\mathrm{c \gt 289}\)
5. INFER the final answer
- The problem states "no real solutions if \(\mathrm{c \gt n}\)"
- We found "no real solutions when \(\mathrm{c \gt 289}\)"
- Therefore, \(\mathrm{n = 289}\)
Answer: 289
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't connect "no real solutions" to the discriminant condition. They might try to solve the quadratic directly or use the quadratic formula without recognizing that the discriminant determines solution existence.
This leads to confusion and guessing since they never establish the proper mathematical framework.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret what "\(\mathrm{c \gt n}\)" means in context. They might think they need to solve for when c equals some specific value rather than finding the threshold where the inequality changes the solution existence.
This may cause them to get stuck after correctly finding \(\mathrm{c \gt 289}\), not realizing that \(\mathrm{n = 289}\).
The Bottom Line:
This problem requires students to shift from thinking about solving equations to thinking about when equations have solutions at all. The key insight is recognizing that discriminant conditions control solution existence, not just solution values.